13 EXTENSIONS OF HERMITIAN OPERATORS 44! and is an eigenvalue of A. The essential spectrum of A is therefore the set of non- isolated points of SpO*). (Reduce to the case A = 0, and by using Problem 1 of Section 15.12 reduce further to the case where Kvr(A) = {0}; then show that 0 is a regular value for A, and use (15.12.11).) (c) If H is a closed hermitian operator of defect (m, 77), then $p(H) contains the half- plane Jz J> 0 (resp. ./z <J 0) if m > 0 (resp. /; > 0). The essential spectrum of H is contained in R. If m and n are finite, and if HI is a closed hermitian operator which extends H, then the essential spectra of H and HI are the same (Section 15.12, Problem l(h)). 12. Let H be an unbounded hermitian operator on a separable Hilbert space E. (a) If Hj, is the restriction of H* to the subspace dom(Ht) = dom(H) -f Ker(#*), show that JEfi is hermitian. (b) Show that if lm(H) is closed in E, then H± is self-adjoint. (If xedomttff), show that H* • x is orthogonal to KerCfiT *), and therefore H* - x e lm(H) (Section 15.12, Problem l(c)); deduce that x e dom(HM If His closed and if there exists a real number which does not belong to the essential spectrum of H, deduce that the defects of H are equal. (c) Show that Efit (resp. E£t) (notation of (15.13.6) is the intersection of Ej (resp. EH) with the orthogonal supplement of Ker(#*). Deduce that if there exists A e R not belonging to the essential spectrum of H, and if the defects of H (which are necessarily equal) have a finite value m, then dim(Ker(jfiT* — A/)) J> m. (d) Suppose that H is closed, and let x be an eigenvector of H* corresponding to a real eigenvalue A. Put x = XQ-\- y-\- z, where XQ e domCfiT), y e E$ and z 6 EH . Show that II^H = ||z|| (reduce to the case A = 0). (e) Suppose that H is closed, and let A be a real number which does not belong to the essential spectrum of H; suppose that the defects of H are equal and finite, say m, and that dim(K&r(H - A/)) is finite, say k. Then dim(Ker(#* - A/)) = m + k. (We may assume that A = 0. By considering the restriction of H to the orthogonal supple- ment of Ker(#), reduce to the case where k = 0. Then deduce from (d) that Ker(7ir*) cannot have dimension >m, by using the hypothesis Kerf-fl"*) n domCfiT) = {0}, and complete the proof by using (c).) 13. Let //"be a closed hermitian operator whose defects are equal and finite, say m. (a) If Vis defined as in (15,13.4), then the self-adjoint extensions A of H are of the form A = /(/ + U)(I— C/)"1, where U is a unitary operator extending V, such that U(Ea) = EH', dom(/4) is therefore the direct sum of domCfiT) and the subspace (/— £/)(£/*) of dimension m, contained in dom(£T*). (b) For a real number A to be an eigenvalue of a self-adjoint extension A of H, it is necessary and sufficient that A should be an eigenvalue of H* (use Problem 12(d)). Show that if A e R is not an eigenvalue of H, then there exists a self-adjoint extension A of H for which A is not an eigenvalue. (Use Problem 12(d) and (e), and choose suitably the unitary operator U of (a).) (c) Suppose that m > 0. For each A 6 R, show that there exists a self-adjoint exten- sion A of H such that A e Sp(^). (Remark that, if A0 is a self-adjoint extension of H and A £ SpC40), then it follows from Problems 12(e) and ll(c) that A is an eigenvalue of H*.) (d) Let A i, A2 be two self-adjoint extensions of H. lfP+ (resp. P~) is the orthogonal projection onto Ejf (resp. E£)> show that the continuous operatoroint